# FLUID PRESSURE

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```					                            VIS UAL PHY SICS
S c hoo l of P hys i cs
U n i ve rs i ty of Sy d ney A u s t r a l i a

FLUID PRESSURE
PASCAL’S PRINCIPLE

?      Why does a brain tumor affect the signal cord?

tumor

Increased
pressure
transmitted down
spinal cord

Pascal's Principle            1653 Blaise Pascal (1623 – 1662)

!     Pressure applied to an enclosed fluid is transmitted
undiminished to every portion of the fluid and walls of the
containing vessel.

ph                                     ph

p0’

p0                                     p0

(0,0)                                 (0,0)
h                                   h
Linear relationship between pressure and depth.
If the pressure at the surface increases then the pressure at
a depth h also increases by the same amount.

ph = p0 + ρ g h            (static fluids only)

a03/p1/fluids/pascal.doc                                                         1
Consequences of Pascal's Principle

• Hydraulic machines - garbage compacters, forklifts,
robots, cherry pickers, plows, tractors, airplane landing
gear, elevators, brake system.
• Water and air beds.
• Glaucoma (eye pressure).
• A sharp blow to the front of an
eyeball will produce a higher
pressure which is transmitted to the
opposite side.
• Bladder empty 150 mmH2O and full 180 mmH2O.
• Measurement of spinal and abdomen pressure.
• Pressure on the abdomen of a pregnant woman is
transmitted to the fetus via the amniotic fluid.
• The motion of soft-bodied animals, like the earthworm,
that have hydrostatic skeletons, using a mesh of
perpendicular muscles a worm squeezes itself into
shape, becoming long and thin or short and fat as
needed.
• Aerosol spray can contains a gas under pressure
called a propellant. It pushes down on the surface of
the liquid that is to be sprayed. When the valve is
opened, the top end of the long tube is at atmospheric
pressure and the bottom end is at a pressure well
above atmospheric pressure. The difference is
pressure propels the liquid up and out.
• Brain tumors increase brain pressure. Also a growing
tumor causes an increased pressure that is transmitted down
the spinal column via the cerebrospinal fluid and may be
detected lower in the spinal cavity that is less invasive than
trying to detect it in the brain itself.

•
→       More examples

a03/p1/fluids/pascal.doc                                                       2
?    How can a person easily lift a car?

F1                             F2

h1

oil

h2

A1                           A2

The same amount of pressure can be produced within a liquid by pistons
of different sizes acting with proportionately different forces with the larger
the cross-sectional area of the piston, the larger the force needed to
create a given pressure. Pascal recognized the tremendous practical
significance of his principle. For the first time since antiquity a new class
of force multipliers known as hydraulic machines (from the Greek for
water and pipe) was possible, although a practical device was not built
until Bramah devised a functioning pressure-seal in 1796. If two
chambers fitted with different-sized pistons are connected so they share a
common working fluid, the pressure generated by one will be transmitted
undiminished to the other. Automobiles are unceremoniously lofted into
the air on hydraulic lifts. Most are activated by compressed air pressing
on oil, but the simplest arrangement is a U-tube, narrow on one side, wide
on the other, with sealed movable pistons at both ends. A downward
input-force F1 acting over the small input-area A1 of the narrow piston
generates an input-pressure p1 = F1 / A1. But this pressure is distributed
uniformly and so equals the output-pressure, p2 = p1 = p which, in turn, is
given by p2 = F2 / A2. Hence F1 / A1 = F2 / A2 and so

F2 / F1 = A2 / A1    F2 = F1 (A2 / A1)        A2 / A1 >> 1 ⇒ F2 >> F1

Although the pressures in and out are equal, the forces are certainly not.
When the output piston under the car has 100 times the area of the input
piston, it will experience an upward force 100 times greater than the
input-force. Like all machines, this is not a work multiplier: at best, when
energy losses (for example, those due to friction) are negligible, work-in
equals work out

a03/p1/fluids/pascal.doc                                                                 3
Work done on piston 1                  W1 = F1 h1 = p A1 h1
Work done by piston 2 in lifting car   W2 = F2 h2 = p A2 h2

W1 = W2 ⇒ p A1 h1 = p A2 h2 ⇒ h1 = h2 (A2 / A1)

One person doing a lot of pumping over a lot of distance h1 can jack a car
up a distance h2 using a relatively small force.

Summary

A1 = 1.00 m2         A2 = 100 m2

⇒

F1 = 200 N         F2 = 20 000 N

h1 = 1.0 m          h2 = 0.010 m = 10 mm

a03/p1/fluids/pascal.doc                                                      4

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